A396594 - OEIS

%I #34 Jul 03 2026 15:58:34

%S 2,3,4,5,7,11,13,17,18,19,23,24,29,31,37,40,41,43,45,47,53,56,59,61,

%T 63,67,71,73,79,83,88,89,97,99,101,103,104,107,109,113,117,127,131,

%U 136,137,139,149,151,152,153,157,163,167,171,173,179,181,184,191,193

%N Numbers m such that rad(m) * bigomega(m) = m.

%C Primes p are terms since rad(p) * bigomega(p) = p * 1 = p.

%C From _Michael De Vlieger_, Jul 03 2026: (Start)

%C Primes are the only squarefree numbers (in A005117) in the sequence. Squarefree k = rad(k) such that bigomega(k) = omega(k) > 2 do not appear in the sequence since k * omega(k) > k.

%C The only powerful number k (in A001694) in this sequence is 4 = 2^2 = 2*2, since powerful numbers k imply rad(k)^2 | k, and further, bigomega(k) >= k/rad(k), where k/rad(k) >= rad(k). Attempting to increment the left hand side only increases the ratio RHS/LHS, since RHS increases by a prime factor.

%C Consequences:

%C 1. A175787 is a proper subset of this sequence.

%C 2. There is no intersection of this sequence and A120944 (squarefree and composite).

%C 3. {a(n)} \ A175787 is a proper subset of A332785.

%C 4. 4 is the only perfect power (in A001597) and the only prime power (in A246547) in this sequence. (End)

%H James C. McMahon, <a href="/A396594/b396594.txt">Table of n, a(n) for n = 1..10000</a>

%e 18 is a term since rad(18) * bigomega(18) = 6 * 3 = 18.

%t q[m_]:=PrimeOmega[m]*Times@@First/@FactorInteger[m]==m;Select[Range[193],q]

%Y Cf. A000040, A001222, A001597, A001694, A007947, A175787, A332785, A397221.

%K nonn

%O 1,1

%A _James C. McMahon_ and _Vincenzo Manto_, Jun 28 2026